Difference between revisions of "Riemann zeta"
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Revision as of 23:07, 6 May 2015
Consider the function $\zeta$ defined by the series $$\zeta(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z}.$$
Graph of $\zeta$ on $[-5,5]$.
- Complex zeta.jpg
Domain coloring of analytic continuation of $\zeta$.
Properties
Proposition: If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges.
Proof: █
Proposition (Euler Product): $\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}}$
Proof: █
Proposition: Let $n$ be a positive integer. Then $$\zeta(2n)=(-1)^{n+1}\dfrac{B_{2n}(2\pi)^{2n}}{2(2n)!},$$ where $B_n$ denotes the Bernoulli numbers.
Proof: █
Theorem
The following formula holds: $$P(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\mu(k)}{k} \log \zeta(kz),$$ where $P$ denotes the Prime zeta function, $\mu$ denotes the Möbius function, $\log$ denotes the logarithm, and $\zeta$ denotes the Riemann zeta function.
Proof
References
External links
- 15 Videos about the Riemann $\zeta$ function
- English translation of Riemann's paper "On the number of prime numbers less than a given quantity"
- Evaluating $\zeta(2)$
- The Riemann Hypothesis: How to make $1 Million Without Getting Out of Bed
- The Riemann Hypothesis: FAQ and resources
- How Euler discovered the zeta function
